REPRESENTATION OF THE HEISENBERG ALGEBRA h₄ BY THE LOWEST LANDAU LEVELS AND THEIR COHERENT STATES

Abstract

Using simultaneous shape invariance with respect to two different parameters, we introduce a pair of appropriate operators which realize shape invariance symmetry for the monomials on a half-axis. It leads to the derivation of rotational symmetry and dynamical symmetry group H4 with infinite-fold degeneracy for the lowest Landau levels. This allows us to represent the Heisenberg–Lie algebra h4 not only by the lowest Landau levels, but also by their corresponding standard coherent states.

On the Generalized Unitary Parasupersymmetry Algebra of Beckers–debergh

“…n and m. On the other hand, the 2D and 3D models representing the algebraic relations (12) for p = 2k are some known quantum mechanical models on the homogeneous manifolds SL(2, c)/GL(1, c) and the group manifolds SL(2, c). Now in order to obtain the mentioned representations, we only introduce two bunches of the shape invariance models which have been classified before [11,25]. In master function theory, a function A(x) which is at most of second order in terms of x, and a non-negative weight function W (x) defined in an interval (a, b) may be chosen so that (1/W (x))(d/dx)(A(x)W (x)) is a polynomial of at most first order.…”

“…In master function theory, a function A(x) which is at most of second order in terms of x, and a non-negative weight function W (x) defined in an interval (a, b) may be chosen so that (1/W (x))(d/dx)(A(x)W (x)) is a polynomial of at most first order. For a given master function A(x) and its corresponding weight function W (x), it has been shown that the eigenvalue equations of the one dimensional partner Hamiltonians corresponding to the first bunch of the superpotentials, which are obtained from the factorization with respect to the main quantum number n, will be [25]…”

“…Meanwhile, by using the definitions (25), the Eqs. (12d) and (12e) lead to the following relations, respectively…”

“…Recently, most of the one dimensional shape invariant solvable quantum mechanical models have been classified into two bunches. The first bunch [25] includes models for which the shape invariance parameter is the main quantum number n. On the other hand, in the second bunch [11] the shape invariance parameter of the models is the secondary quantum number m. Meanwhile, it has been shown in Ref. [11] that the R-S parasupersymmetry algebra of arbitrary order p is realized by the shape invariant quantum mechanical models so that the algebra can be represented by the quantum mechanical states of the models.…”